4. Measuring Chaos

4.2 Logistic Equation

The simple logistic equation is a formula for
approximating the evolution of an animal population over time. Many animal species
are fertile only for a brief period during the year and the young are born in
a particular season so that by the time they are ready to eat solid food it will
be plentiful. For this reason, the system might be better described by a discrete
difference equation than a continuous differential equation.

Since not every existing
animal will reproduce (a portion of them are male after all), not every female
will be fertile, not every conception will be successful, and not every pregnancy
will be successfully carried to term; the population increase will be some fraction
of the present population. Therefore, if "An" is the number
of animals this year and "An+1" is the number next year,
then

An+1 = rAn

where "r" is the growth rate or fecundity,
will approximate the rate of succesful reproduction.

This model produces exponential
growth without limit. Since every population is bound by the physical limitations
of its territory, some allowance must be made to restrict this growth. If there
is a carrying-capacity of the environment then the population may not exceed that
capacity. If it does, the population would become extinct. This can be modeled
by multiplying the population by a number that approaches zero as the population
approaches its limit. If we normalize the "An" to this capacity
then the multiplier (1 − An) will suffice and the resulting
logistic equation becomes

An+1 = rAn(1 − An)

or in functional form

ƒ(x) = rx (1 − x).

The logistic equation is parabolic like the quadratic mapping with ƒ(0) = ƒ(1) = 0
and a maximum of ¼r at ½. Varying the parameter changes the height of
the parabola but leaves the width unchanged. (This is different from the quadratic
mapping which kept its overall shape and shifted up or down.) The behavior of
the system is determined by following the orbit of the initial seed value. All
initial conditions eventually settle into one of three different types of behavior.

Fixed: The population approaches a stable value. It can do so by approaching asymptotically from one side in a manner something like an over damped harmonic oscillator or asymptotically from both sides like an under damped oscillator. Starting on a seed that is a fixed point is something like starting an SHO at equilibrium with a velocity of zero. The logistic equation differs from the SHO in the existence of eventually fixed points. It's impossible for an SHO to arrive at its equilibrium position in a finite amount of time (although it will get arbitrarily close to it).

Periodic: The population alternates between two or more fixed values. Likewise, it can do so by approaching asymptotically in one direction or from opposite sides in an alternating manner. The nature of periodicity is richer in the logistic equation than the SHO. For one thing, periodic orbits can be either stable or unstable. An SHO would never settle in to a periodic state unless driven there. In the case of the damped oscillator, the system was leaving the periodic state for the comfort of equilibrium. Second, a periodic state with multiple maxima and/or minima can arise only from systems of coupled SHOs (connected or compound pendulums, for example, or vibrations in continuous media). Lastly, the periodicity is discrete; that is, there are no intermediate values.

Chaotic: The population will eventually visit every neighborhood in a subinterval of (0, 1). Nested among the points it does visit, there is a countably infinite set of fixed points and periodic points of every period. The points are equivalent to a Cantor middle thirds set and are wildly unstable. It is highly likely that any real population would ever begin with one of these values. In addition, chaotic orbits exhibit sensitive dependence on initial conditions such that any two nearby points will eventually diverge in their orbits to any arbitrary separation one chooses.

The behavior of the logistic equation is more complex than that of the simple
harmonic oscillator. The type of orbit depends on the growth rate parameter, but
in a manner that does not lend itself to "less than", "greater
than", "equal to" statements. The best way to visualize the behavior
of the orbits as a function of the growth rate is with a bifurcation diagram.
Pick a convenient seed value, generate a large number of iterations, discard the
first few and plot the rest as a function of the growth factor. For parameter
values where the orbit is fixed, the bifurcation diagram will reduce to a single
line; for periodic values, a series of lines; and for chaotic values, a gray wash
of dots.

Since the first two chapters of this work were filled will bifurcation diagrams
and commentary on them, I won't go much into the structure of the diagram other
than to locate the most prominent features. There are two fixed points for this
function: 0 and 1 − 1/r, the former being stable on the interval (−1, +1)
and the latter on (1, 3). A stable 2-cycle begins at r = 3 followed
by a stable 4-cycle at r = 1 + √6. The period continues
doubling over ever shorter intervals until around r = 3.5699457… where
the chaotic regime takes over. Within the chaotic regime there are interspersed
various windows with periods other than powers of 2, most notably a large 3-cycle
window beginning at r = 1 + √8. When the growth rate
exceeds 4, all orbits zoom to infinity and the modeling aspects of this function
become useless.

The first chapter introduces the basics of one-dimensional iterated maps. Say what? Take a function y = ƒ(x). Substitute some number into it. Take the answer and run it through the function again. Keep doing this forever. This is called iteration. The numbers generated exhibit three types of behavior: steady-state, periodic, and chaotic. In the 1970s, a whole new branch of mathematics arose from the simple experiments described in this chapter.

The second chapter extends the idea of an iterated map into two dimensions, three dimensions, and complex numbers. This leads to the creation of mathematical monsters called fractals. A fractal is a geometric pattern exhibiting an infinite level of repeating, self-similar detail that can't be described with classical geometry. They are quite interesting to look at and have captured a lot of attention. This chapter describes the methods for constructing some of them.

The third chapter deals with some of the definitions and applications of the word dimension. A fractal is an object with a fractional dimension. Well, not exactly, but close enough for now. What does this mean? The answer lies in the many definitions of dimension.

The fourth chapter compares linear and non-linear dynamics. The harmonic oscillator is a continuous, first-order, differential equation used to model physical systems. The logistic equation is a discrete, second-order, difference equation used to model animal populations. So similar and yet so alike. The harmonic oscillator is quite well behaved. The paramenters of the system determine what it does. The logistic equation is unruly. It jumps from order to chaos without warning. A parameter that discriminates among these behaviors would enable us to measure chaos.